AN UPPER BOUND FROM HELIOSEISMOLOGY ON THE STOCHASTIC BACKGROUND OF GRAVITATIONAL WAVES

A stochastic background of gravitational waves (SBGW) is expected to result from the incoherent superposition of gravitational radiation from a large number of sources (Maggiore 2000; Sathyaprakash & Schutz 2009). It can be subdivided into a cosmological background of primordial gravitational waves (GWs) generated by processes of the very early universe, and into an astrophysical background produced by a large number of unresolved astrophysical sources that consist of an accelerated mass distribution with a quadrupole moment. Many potential processes of the very early universe have been proposed as generators of cosmological SBGWs, examples of which are standard inflationary models, pre-Big-Bang models, and cosmic strings (Maggiore 2000). Candidate sources to produce the astrophysical component are, for instance, compact binary star systems, core collapse supernovae, and rotating neutron stars (Schneider et al. 2010; Regimbau 2011; Amaro-Seoane et al. 2013). Since the universe has been essentially transparent to gravitational radiation from the very beginning, GWs are ideal carriers of information on the physical processes that generated them and thus on the state of the universe at the time they were produced. This background encodes information on the very early epoch as well as on the more recent history of the universe that is not accessible to conventional astronomical observations based on electromagnetic waves (Maggiore 2000; Sathyaprakash & Schutz 2009; Schneider et al. 2010; Hils et al. 1990). Extracting this astrophysical and cosmological information from the SBGW and thus disentangling the various contributions requires GW experiments in a variety of frequency bands.

Tight upper bounds on the SBGW have been placed at high and low frequencies by Earth-based interferometric detectors such as LIGO (around 100 Hz; Abbott et al. 2009), pulsar timing observations based on the residuals of pulse arrival times (between 10⁻⁹ − 3 × 10⁻⁸ Hz; Jenet et al. 2006), and cosmic microwave background (CMB) observations at large angular scales which indicate an upper limit on the cosmological component of the SBGW at larger wavelengths than the horizon size at the time the CMB was produced (3 × 10⁻¹⁸–10⁻¹⁶ Hz; Maggiore 2000). However, the intermediate frequency regime between 10⁻⁸–10 Hz has proven to be particularly challenging to be probed by GW experiments and has thus remained essentially unexplored until today. Apart from the frequency-independent indirect limits from CMB and Big-Bang nucleosynthesis (BBN) data (Smith et al. 2006; Cyburt et al. 2005), which only place tight bounds on the cosmological component of the SBGW, there are limits from Doppler tracking of the Cassini spacecraft (10⁻⁶–10⁻³ Hz; Armstrong et al. 2003), from precision orbital monitoring of the Hulse–Taylor binary pulsar (at 10⁻⁴ Hz; Hui et al. 2013), from seismic data of the Earth (0.05–1 Hz; Coughlin & Harms 2014), and from a pair of torsion-bar antennas (TOBAs; 0.035–0.830 Hz; Shoda et al. 2014). In the 10⁻⁴–10 Hz regime, in particular, the astrophysical component of the SBGW could well outshine the cosmological component, revealing rich astrophysical information on, for example, the physics of compact objects and star formation history (Regimbau 2011; Schneider et al. 2010; Hils et al. 1990). Proposed space missions to explore the latter frequency range are the New Gravitational wave Observatory NGO (a.k.a. eLISA; Amaro-Seoane et al. 2013) and the DECi-hertz Interferometer Gravitational wave Observatory DECIGO (Kawamura et al. 2011).

Recent theoretical work on the excitation of global stellar oscillations by GWs (Siegel & Roth 2010, 2011) together with helio- and asteroseismic data allows us to employ the Sun and other solar-like stars as astronomical test masses for the detection of GWs. Helioseismic Sun-as-a-star data for internal gravity and pressure modes of the Sun is available from, e.g., the Global Oscillation at Low Frequency (GOLF) instrument aboard the Solar and Heliospheric Observatory (SOHO) spacecraft (Gabriel et al. 1995), while asteroseismic data from missions like CoRoT (Baglin et al. 2006) and Kepler (Christensen-Dalsgaard et al. 2009) offer the possibility to employ other solar-like stars.

Here, we propose a method to directly detect or constrain both the astrophysical and cosmological component of the SBGW at mHz and μHz frequencies by using asteroseismic observations (Section 2). While this method is general and can be applied to any star, we demonstrate the feasibility of this approach by deducing a direct upper bound around 0.11 mHz using our Sun as a hydrodynamic detector (Section 3). Section 4 is devoted to discussion and conclusions.

Our method is based on the theoretical result that stellar oscillations can be excited by an SBGW (Siegel & Roth 2010, 2011). In the case of the Sun, the resulting amplitudes can be close to or comparable with values expected from excitation by near-surface convection (Siegel & Roth 2011), which is considered to be the main driving mechanism for oscillations in the Sun and solar-like stars (e.g., Goldreich & Keeley 1977; Balmforth 1992; Goldreich et al. 1994; Samadi & Goupil 2001; Belkacem et al. 2008). The physical mechanism underlying the excitation by an SBGW is the fact that GWs manifest themselves in oscillating tidal forces: they periodically stretch and compress the spatial dimensions orthogonal to the direction of propagation in a quadrupolar pattern,³ thus imposing stresses on the matter they pass through. Global stellar oscillations can be classified into acoustic pressure modes (p modes), with pressure gradients as the restoring force, and interior gravity modes (g modes), with buoyancy as the restoring force (Aerts et al. 2010; Unno et al. 1989). Both p and g modes can be excited by GWs, although only quadrupolar eigenmodes can attain non-vanishing amplitudes due to the quadrupole nature of the excitation force. This fact can help to distinguish an SBGW from other excitation mechanisms (the "noise").

The equation of motion for stellar oscillations in presence of external driving by GWs is the inhomogeneous wave equation for the velocity field ${\boldsymbol{v}}$ (cf. Siegel & Roth 2011)

$$\begin{equation} \rho \left(\frac{\partial ^2}{\partial t^2}-\mathcal {L}\right){\boldsymbol{v}}+\mathcal {D} ({\boldsymbol{v}})=\frac{\partial }{\partial t} \, ({\boldsymbol{f}}_{\rm {Rey}}+{\boldsymbol{f}}_{\rm {entr}}+{\boldsymbol{f}}_{\rm {GW}}), \end{equation} \tag{ 1 }$$

where ρ is the density and $\mathcal {L}$ and $\mathcal {D}$ are linear differential operators (see Siegel & Roth 2011 for definitions). Furthermore, ${\boldsymbol{f}}_{\rm {Rey}}$ and ${\boldsymbol{f}}_{\rm {entr}}$ denote the Reynold and entropy source terms, respectively, and represent the driving terms due to convective motions (see Siegel & Roth 2011 and Samadi & Goupil 2001 for definitions). In the following, we are solely interested in the driving term due to GWs, ${\boldsymbol{f}}_{\rm {GW}}$, with components given by

$$\begin{equation} f_{\rm {GW}}^i({\boldsymbol{x}},t)=\frac{1}{2}\rho x^j \frac{\partial ^2 }{\partial t^2} h^i_{\phantom{i}j}. \end{equation} \tag{ 2 }$$

Here, h_μν denotes the GW tensor.⁴

The velocity field in Equation (1) can be expanded in terms of the complete set of eigenfunctions $\lbrace {\boldsymbol{\xi }}_N({\boldsymbol{x}})\rbrace$ of the operator $\mathcal {L}$, which define the solutions

$$\begin{equation} {\boldsymbol{v}}_{{\mathrm{hom}}}({\boldsymbol{x}},t)=-{\mathrm{i}}\omega {\boldsymbol{\xi }}({\boldsymbol{x}}){\mathrm{e}}^{-{\mathrm{i}}\omega t} \end{equation} \tag{ 3 }$$

of the homogeneous problem

$$\begin{equation} \left(\frac{\partial ^2}{\partial t^2}-\mathcal {L}\right){\boldsymbol{v}}_{\rm {hom}}=0, \end{equation} \tag{ 4 }$$

$$\begin{equation} {\boldsymbol{v}}({\boldsymbol{x}},t)=\sum _N (-{\mathrm{i}}\omega _N) A_N(t){\boldsymbol{\xi }}_N({\boldsymbol{x}}){\mathrm{e}}^{-{\mathrm{i}}\omega _N t}. \end{equation} \tag{ 5 }$$

Hence, the intrinsic complex velocity field associated with a stellar oscillation normal mode as a function of time t and position $\boldsymbol {x}$ within the star can be written as

$$\begin{equation} \boldsymbol {v}_N(\boldsymbol {x},t)\equiv -i\omega _N A_N(t)\boldsymbol {\xi }_N(\boldsymbol {x})e^{-i\omega _N t}, \end{equation} \tag{ 6 }$$

where A_N(t) is a time-dependent complex amplitude, $\boldsymbol {\xi }_N(\boldsymbol {x})$ denotes the displacement eigenfunction, ω_N the oscillation frequency, and N = (nlm) is an abridged index denoting the eigenmode under consideration with radial order n, harmonic degree l and azimuthal order m. In the pulsation frame of a slowly rotating star (polar axis coincides with rotation axis) and spherical coordinates, the displacement eigenfunction can be written as

$$\begin{equation} \boldsymbol {\xi }_N(r,\Theta,\Phi)=[\xi _{r,nl}(r) \boldsymbol {e}_r +\xi _{h,nl}(r)r\nabla ]Y_{lm}(\Theta,\Phi), \end{equation} \tag{ 7 }$$

which we normalize according to

$$\begin{equation} \int _V d^3x\,\rho \boldsymbol {\xi }_N^*\boldsymbol {\,{\cdot}\, \xi }_{N^{\prime }}=I\delta _{NN^{\prime }}. \end{equation} \tag{ 8 }$$

In the above equations, * denotes complex conjugation, V the stellar volume, I is a constant that we set to unit mass in cgs units, ξ_{r, nl}(r) and ξ_{h, nl}(r) denote the radial and horizontal eigenfunctions, respectively, and Y_lm(Θ, Φ) is the spherical harmonic associated with the oscillation mode.

The quantity that is directly accessible to asteroseismic and Sun-as-a-star radial velocity measurements is the disk-integrated, apparent surface velocity of a mode (the observed root mean square (rms) surface velocity), which can be expressed as (cf. also Belkacem et al. 2009; Berthomieu & Provost 1990)

$$\begin{equation} v_N=\left\langle \frac{1}{2}\vert v_{{\rm app},N}(R_0,t)\vert ^2\right\rangle ^{1/2}, \end{equation} \tag{ 9 }$$

where 〈〉 denotes temporal average and where we defined the complex apparent surface velocity

$$\begin{equation} v_{{\rm app},N}(R_0,t)=\frac{\displaystyle \int _H h(\mu)\boldsymbol {v}_N(\boldsymbol {x},t)\boldsymbol {\,{\cdot}\, n}\,d\Omega }{\displaystyle \int _H h(\mu)\,d\Omega } \end{equation} \tag{ 10 }$$

as observed in the observer's frame with spherical coordinates (r, θ, ϕ) (the direction θ = 0 points toward the observer and r = 0 corresponds to the center of the star). In Equation (10), H denotes the visible half sphere corresponding to the parameter space {(θ, ϕ)|0 < θ < π/2, 0 < ϕ < 2π}, μ = cos θ the limb angle, $d\Omega =d\boldsymbol {A\cdot n}=R_0^2\sin \theta \cos \theta d\theta d\phi$ the surface element projected onto the direction of the observer, and R₀ the distance between the center of the star and the layer where the apparent surface velocity is observed. Furthermore, $\boldsymbol {n}=\cos \theta \boldsymbol {e}_r-\sin \theta \boldsymbol {e}_\theta$ is the unit vector at a particular position on H, pointing toward the observer, and h(μ) is an appropriate limb darkening law. In other words, the apparent surface velocity v_N is the intrinsic line-of-sight velocity at a certain layer in the atmosphere, integrated and weighted over the visible stellar disk according to a certain limb darkening function.

In the case of excitation by an SBGW (i.e., ${\boldsymbol{f}}_{\rm {Rey}}={\boldsymbol{f}}_{\rm {entr}}\equiv 0$), the intrinsic mean-square amplitude of a quadrupolar (l = 2) stellar oscillation mode is time independent and can be directly expressed in terms of the normalized spectral energy density of the background as (Siegel & Roth 2011):

$$\begin{equation} \langle |A_N|^2\rangle =\frac{\pi ^2}{25}\frac{\chi _n^2}{\eta _N\omega _N I^2}H_0^2\Omega _{\mathrm{GW}}(\omega _N), \end{equation} \tag{ 11 }$$

where

$$\begin{equation} \chi _n=\int _0^R \rho (r) r^3 [\xi _{r,n2}(r)+3\xi _{h,n2}(r)]\,dr \end{equation} \tag{ 12 }$$

is part of the coupling factor between the GW field and the stellar oscillation mode. Here, R denotes the stellar radius and η_N is the damping rate associated with mode N. The normalized dimensionless function

$$\begin{equation} \Omega _{\mathrm{GW}}(\nu)\equiv \frac{1}{\rho _{\mathrm{crit}}}\frac{d\rho _{\mathrm{GW}}}{d\ln \nu } \end{equation} \tag{ 13 }$$

is a convenient way of characterizing the properties of an SBGW (Maggiore 2000; Allen & Romano 1999). It measures the energy density of GWs per unit logarithmic frequency interval in units of the present critical energy density, $\rho _{\mathrm{crit}}=3c^2H_0^2/8\pi G$, that is needed for a closed geometry of the universe. Here, H₀ denotes the present Hubble expansion rate, c the speed of light, and G the gravitational constant.

Furthermore, it can be shown that the intrinsic amplitude A_N is related to v_N as

$$\begin{equation} v_N=\frac{1}{\sqrt{2}}\langle |A_N(t)|^2\rangle ^{1/2} \omega _N \Psi _N(R_0), \end{equation} \tag{ 14 }$$

where

$$\begin{equation} \Psi _N(R_0)=|\alpha _{lm} \xi _{r,nl}(R_0)+\beta _{lm} \xi _{h,nl}(R_0)|, \end{equation} \tag{ 15 }$$

with visibility coefficients

$$\begin{eqnarray} \alpha _{lm} &=& N_{lm}|P_{lm}(\cos \Theta _0)|u_l, \end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray} \beta _{lm} &=& N_{lm}|P_{lm}(\cos \Theta _0)|v_l \end{eqnarray} \tag{ 17 }$$

(cf. also Belkacem et al. 2009; Berthomieu & Provost 1990; Dziembowski 1977; Christensen-Dalsgaard & Gough 1982). Here, $N_{lm}=\sqrt{(2l+1)/4\pi }\sqrt{(l-m)!/(l+m)!}$, P_lm denote the associated Legendre polynomials, and

$$\begin{eqnarray} u_l&=&\int _{0}^{1} \tilde{h}(\mu)\mu ^2 P_l(\mu)\,d\mu, \quad \qquad \qquad \end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray} v_l&=&l\int _{0}^{1} \tilde{h}(\mu)\mu [P_{l-1}(\mu)-\mu P_l(\mu)]\,d\mu, \end{eqnarray} \tag{ 19 }$$

with $\tilde{h}(\mu)=h(\mu)/\int _0^1 h(\mu)\mu \,d\mu$ and P_l = P_l0 the Legendre polynomials.

Given an upper bound on the apparent surface velocity v_N of a quadrupolar stellar eigenmode with frequency ω_N = 2πν_N and assuming that the observed oscillations of a star are excited at least partially by a stochastic background of gravitational radiation as one of the driving forces, Equations (11) and (14) imply an upper limit on the normalized spectral energy density of the SBGW according to

$$\begin{equation} H_0^2\Omega _{\mathrm{GW}}(\nu _N)<\frac{25}{\pi ^3}\mathcal {X}_N\mathcal {M}_N\frac{\eta _N}{\nu _N}v_N^2. \end{equation} \tag{ 20 }$$

Here, $\mathcal {X}_N\equiv 1/\chi ^2_n$ is the coupling factor between the GW field and the stellar oscillation mode, i.e., the susceptibility of a particular mode to the GW background. The factor $\mathcal {M}_N\equiv I^2/\Psi _N^2(R_0)$ is the observed mode mass (the observed value for the total interior mass of the star that is affected by the oscillation), which takes instrumental and other observation-related effects into account (cf. Equation (15)). We note that the left-hand side of Equation (20) is independent of the Hubble constant and thus independent of its uncertainty.

The method described in the previous section is very general and applies to any solar-like oscillator. Here, we compute an upper limit employing our nearest star, the Sun, as a hydrodynamic detector. The underlying solar model that we use is Model S (Christensen-Dalsgaard et al. 1996), which is extensively used as a reference solar model (see, e.g., Turck-Chièze & Couvidat 2011 for a discussion on the current status of the standard solar model). Thanks to the very high level of agreement in the numerical results for solar(-like) models computed with present stellar evolution codes (Lebreton et al. 2008), any other well-fitted solar model could have been used, such as, e.g., the CESAM model employed by Belkacem et al. (2009). We note that the mean quadratic differences in the physical and seismic variables between solar models computed with, e.g., ASTEC and CESAM are particularly small (mean quadratic differences in the physical and thermodynamic variables are often well below or on the order of 1%; differences in the oscillation frequencies are typically less than 0.01%; Lebreton et al. 2008).

The solar oscillation modes best-suited for deducing an upper bound on a GW background are the high-frequency (low radial order) quadrupolar g modes, as they are most sensitive to such a background (Siegel & Roth 2011): the corresponding intrinsic mean-square amplitudes 〈|A_N|²〉 are typically orders of magnitude larger than for low radial order quadrupolar p modes (assuming a constant Ω_GW(ν)), which is mainly the result of much smaller damping rates (cf. Figure 5 in Siegel & Roth 2011). Combined with the observed much larger surface amplitudes, p modes are less relevant for deducing a tight upper bound on an SBGW and are thus not considered here (note that the upper bound on an SBGW according to Equation (20) scales with the squared surface velocity).

Solar g modes have not been unambiguously detected so far, although some g-mode candidates have been identified in data from the GOLF instrument aboard the SOHO spacecraft (Turck-Chièze et al. 2004; García et al. 2007; for a review on quadrupolar p and g-mode measurements, see, e.g., Turck-Chièze & Lopes 2012). Their surface velocities are extremely small, because these modes are evanescent in the convection zone. Theoretical quantitative estimates for solar g-mode surface velocities are highly uncertain and differ from each other by orders of magnitude (e.g., Gough 1985; Bahcall & Kumar 1993; Kumar et al. 1996; Provost et al. 2000; Belkacem et al. 2009), which is predominantly due to the hypotheses made in the excitation models concerning turbulent convection and, in particular, the choice of the eddy-time correlation function (Belkacem et al. 2009). The currently predicted range for quadrupolar g-mode rms surface velocities is 10⁻³ mm s⁻¹ ≲ v_N ≲ 1 mm s⁻¹ (cf. Appourchaux et al. 2010).

Observational upper bounds on g-mode surface amplitudes have been reported numerously in the literature (Appourchaux et al. 2010). As a conservative upper bound on solar g-mode rms surface amplitudes we adopt 6 mm s⁻¹ for any of the modes (Gabriel et al. 2002), which is the 90% confidence limit deduced from Doppler measurements of the solar disk-integrated line-of-sight velocity field as observed by the GOLF instrument. This limit is based on Level 2 GOLF data, which were calibrated using the method described by Ulrich et al. (2000). Accordingly, we employ for our calculation of visibility coefficients the limb darkening law found by Ulrich et al. (2000),

$$\begin{equation} h(\mu)=1+c_1(1-\mu)+c_2(1-\mu)^2+c_3(1-\mu)^3, \end{equation} \tag{ 21 }$$

where c₁ = −0.466, c₂ = −0.06, and c₃ = −0.29. With this limb darkening law, the numerical values for the coefficients u₂ and v₂ (cf. Equations (18) and (19)) are 0.324 and 0.781, respectively. We numerically checked that the precise height in the solar atmosphere where the velocities are observed does not significantly influence our final result. To a very good approximation, we can therefore set R₀ = R, where R is the stellar radius. This is to be expected for g modes, given their small amplitudes at the solar surface. The angle between the solar rotation axis and the polar axis of the observer's frame as defined by the position of the SOHO spacecraft is Θ₀ = 83°. It is worthwhile to note that the visibility coefficients do not depend on the (time-dependent) azimuthal offset Φ₀ between the pulsation frame and the observer's frame. With these details, the values of the visibility coefficients α_lm and β_lm (cf. Equations (16) and (17)) can be computed, which we list in Table 1.

A further ingredient are the damping rates. It is important to note that radiative damping is the dominant damping mechanism for asymptotic g modes. Thanks to this fact, g-mode damping rates (excluding the first few radial orders, i.e., at least up to ≈110 μHz in the case of the Sun) can be reliably calculated with non-adiabatic oscillation computations (Dupret 2002; Belkacem et al. 2009). Above 110 μHz, time-dependent convection terms become significant and damping rates are sensitive to the parameter β (cf. Grigahcène et al. 2005) used by Belkacem et al. (2009) to model convection-pulsation interactions. The value adopted by the latter authors was chosen such that good agreement between theoretically computed and observed damping rates of solar p modes was achieved. However, for the lack of corresponding data, this calibration cannot be verified for solar g modes. Although the deviation from the radiative damping power law (see below) will most likely not be large for the first few modes above 110 μHz, we cannot entirely trust the upper bounds deduced above this frequency.

Figure 1 shows the upper bounds on an SBGW as deduced from quadrupolar (l = 2) asymptotic solar g modes according to Equation (20), in which we have assumed H₀ = 70 km s⁻¹ Mpc⁻¹ (Komatsu et al. 2011). As can be seen from Figure 1, the upper bounds are tightest at high frequencies and they follow a fairly regular power law, which can be understood from an asymptotic analysis of the quantities appearing in Equation (20). The quantities $\chi _n^2$ and η_N are known to show power-law behavior as a function of frequency for asymptotic g modes (Siegel & Roth 2011; Belkacem et al. 2009). Here, we also find a power law for $\Psi _N^2(R_0)$ in the case of asymptotic g modes. Therefore, according to Equation (20), the upper bound on Ω_GW(ν) is also of power-law form. We note that the degeneracy in the azimuthal order m of the modes is lifted due to visibility effects, which are encoded in the quantity $\Psi _N^2(R_0)$ through the visibility coefficients α_lm and β_lm. The regime above ≈110 μHz, in which we cannot entirely trust the damping rates (see above), is indicated in gray. The tightest bound that can still be reliably deduced is Ω_GW < 1.7 × 10⁸ at 0.112 mHz, while the tightest overall bound is Ω_GW < 4.0 × 10⁵ at 0.171 mHz. These values are reduced by a factor of four if an upper bound of 3 mm s⁻¹ on the apparent g-mode surface velocities according to Turck-Chièze et al. (2004) is used.⁵ As evident from Figure 1, the tightest bounds are obtained for g modes with l = m = 2, which are compared to other observational constraints on an SBGW in Figure 2. In the latter figure, we mark the former of the aforementioned limits with a large dot.

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The frequency range 10⁻⁴–10 Hz, which is particularly interesting from the astrophysical point of view, has essentially remained unexplored until today in terms of strong bounds. However, several low-frequency antennas are currently being proposed, such as NGO (a.k.a. eLISA), DECIGO, or TOBA, with final sensitivities that are claimed to reach Ω_GW ∼ 10⁻⁹, 10⁻¹⁵ and Ω_GW ∼ 10⁻⁸, respectively (Amaro-Seoane et al. 2013; Kawamura et al. 2011; Ishidoshiro et al. 2011). In this frequency range, cosmological backgrounds (e.g., from cosmic strings) could be outshined by the astrophysical backgrounds from binary neutron stars and galactic as well as extragalactic white dwarfs binaries (Regimbau 2011; Schneider et al. 2010; Amaro-Seoane et al. 2013). At lower frequencies where our method can still be applied, the astrophysical background due to supermassive black hole binaries becomes important (e.g., Sesana et al. 2004). The method presented in this paper makes the aforementioned frequency range accessible by providing a possibility to place direct bounds on an SBGW at μHz and mHz frequencies with asteroseismic data. Despite some partial overlap with the Cassini band, this method can somewhat bridge the gap between the Cassini range and the bounds from Earth-based interferometers, possibly improving on the Cassini limits (see below). It is important to point out that there are indirect upper limits in this frequency range, which are deduced from CMB and BBN data due to the fact that a larger amplitude of the SBGW would have altered the observed abundances of the light nuclei created during BBN; analogously, a larger amplitude would have also modified the CMB and matter power spectra. However, these are indirect bounds in the sense that they constrain the integrated total energy density

$$\begin{equation} \Omega _\mathrm{GW}=\int \Omega _{\mathrm{GW}}(\nu) d\ln \nu \end{equation} \tag{ 22 }$$

over the frequency ranges indicated in Figure 2 as Ω_{GW, BBN} < 1.6 × 10⁻⁵ and Ω_{GW, CMB} < 1.7 × 10⁻⁵, respectively (Maggiore 2000; Cyburt et al. 2005; Smith et al. 2006). Furthermore, they solely constrain primordial GWs, which already existed at the time when, respectively, the CMB was formed and the nucleosynthesis took place. In particular, these bounds cannot constrain the astrophysically generated GWs, which were produced at later times and which are still generated today. In contrast to these indirect limits, our method reported here yields direct upper bounds that additionally constrain the astrophysical component of the stochastic GW background. They complement the direct upper limits in other frequency bands and the integrated, indirect upper limits in the same frequency regime. Furthermore, unlike most of the other direct bounds, our method does not assume a certain spectral shape for the function Ω_GW(ν), i.e., it applies to an arbitrary function Ω_GW(ν), regardless of the global shape of Ω_GW(ν).

The upper bound given by Equation (20) is highly sensitive to local (i.e., seismic; cf. the wide range in sensitivity in Figures 1 and 2) and global stellar properties and to the observed surface velocities of the modes. Space missions like CoRoT and Kepler recorded asteroseismic intensity data for a wide range of stellar mass, radius and effective temperature. Furthermore, radial velocity measurements of stellar oscillations from ground exist. The precision of helioseismic measurements might not be achieved by asteroseismology. In particular, from solar observations it can be concluded that the signal-to-noise ratio of intensity variations are expected to be an order of magnitude lower than for velocity measurements (Nigam et al. 1998), and radial velocity measurements might not yet achieve a precision on the order of mm/s. However, these limitations could be overcompensated by several orders of magnitude due to the dependence of Equation (20) on, e.g., stellar mass and radius (Siegel & Roth 2011), and stellar modelling can guide observations to find optimal targets. Still, a conversion from intensity to velocity amplitudes is required before using intensity data in Equation (20), but this might be, e.g., obtained empirically from comparisons of intensity and velocity measurements, as, for instance, carried out by SOHO for the Sun (Nigam et al. 1998) or between Kepler intensity and follow-up radial velocity data for the stars. Therefore, the existing and up-coming space-based missions, as well as ground-based facilities like SONG (Grundahl et al. 2009), offer a unique possibility to focus on targets with optimized sets of stellar parameters in terms of detector sensitivity and to possibly employ these stars as a large array of low-frequency antennas for the SBGW at μHz and mHz frequencies by using the method presented here.

The authors thank K. Belkacem and R. Samadi for valuable discussions and K. Belkacem for sharing the solar g-mode damping rates published in Belkacem et al. (2009).